System and method for transmitting picture or sound signals

ABSTRACT

Due to a reduction of block artefacts in the reconstructed signal, lapped signal transforms are more suitable than non-lapped transforms for compressing picture and sound signals. Nevertheless, the known lapped orthogonal transform is still accompanied by the artefacts. A system and method are described for transmitting picture or sound signals in which the artefacts remain absent also upon substantial compression and in which orthogonality and phase linearity of the transform is maintained. The transform matrix may be constructed from any arbitrary orthogonal K*K basis matrix C. It is indicated how the basis matrix can be obtained, for example from the DCT.

BACKGROUND OF THE INVENTION

The invention relates to a system for transmitting picture or soundsignals. More particularly, the invention relates to transmitting thesesignals by means of a lapped transform method.

In the lapped transform of a picture or sound signal, series of 2Ksamples of the signal halfway overlapping each other are transformedinto K coefficients. The coefficients are transmitted and subsequentlybackward-transformed to reconstruct the picture or sound signal. Thetransform is described by a transform matrix. The transform matrix atthe transmitter end will hereinafter be denoted by A. The transformmatrix at the receiver end will hereinafter be denoted by S. Thematrices A and S have a dimension of K*2K (vertical*horizontal)elements. The K rows of 2K elements of the matrix S constitute the basisfunctions of the transform. Each basis function has a frequencyspectrum. The basis functions are chosen to be such that each basisfunction comprises a part of the total frequency spectrum. As thecentral frequency of that part is higher, the basis function is said tohave a higher order. For example, in picture transforms an increasingorder represents an increasing extent of picture detail.

Signal transforms are often used for digital compression of picture andsound signals. Compression is possible because coefficients of a higherorder are often coarsely quantized or may even be omitted. It isimportant that the basis functions of the transform are chosencarefully. A very frequently used transform method for compressing videopictures is the Discrete Cosine Transform (DCT). This is a non-lappedtransform whose matrix consists alternately of even and odd basisfunctions of an increasing order.

A known lapped transform method is published in "Lapped Transforms forEfficient Transform/Sub-band Coding", IEEE Trans. on ASSP, vol. 38, no.6, June 1990, pp. 969-978. The transform matrices A and S are derivedfrom the non-lapped DCT. They comprise half a number of even rows andhalf a number of odd rows. The even rows are obtained by pair-wiseforming the difference of an even row and the subsequent odd row of theDCT matrix and by repeating said difference in a mirrored fashion. Theodd rows are obtained by repeating the same difference negatively and ina mirrored fashion. The matrices may thus be mathematically written as:##EQU1## In this formula, I is the unit matrix, O is the zero matrix andJ is the counter-identity matrix. C_(c) and C_(o) are 1/2K*K matrices inwhich the even and odd rows of a matrix C are accommodated. T is amatrix which ensures that the odd rows do not show largediscontinuities. The rows of C are denoted by c_(j), in which the indexj represents the order of the relevant basis function. An even index jalso indicates that the basis function is even. An odd index j indicatesthat the basis function is odd.

In accordance with the state of the art, the K*K matrix C is constitutedby the DCT matrix. The state of the art thus indicates a method ofconstructing a lapped transform from the non-lapped DCT. The DCT matrixfunctions, as it were, as a "basis matrix" for the construction of A andS. The lapped transform obtained is orthogonal. The transform istherefore also referred to as LOT (Lapped Orthogonal Transform).Orthogonality is a desired property in signal transforms due to thepreservation of energy and because the transmission system is thenwell-conditioned. This is understood to mean that the system behaves ina numerically stable way. Moreover, in orthogonal transforms, thebackward transform matrix S is identical to the transmitter matrix A.The known lapped transform is also phase-linear. Phase linearity meansthat the delay time of the transmission system is the same for allpicture or sound frequencies.

However, the known LOT has the drawback that discontinuities occur atthe edges of each series of samples of the output signal when thecoefficients are quantized for compression (i.e. data reduction). Inpicture coding, in which the picture is divided into 2-dimensionalblocks, these errors become manifest as visible luminancediscontinuities at the edges of each block. These discontinuities aretherefore also referred to as block artefacts.

SUMMARY OF THE INVENTION

It is an object of the invention to provide a system for transmittingpicture and sound signals in which said artefacts are substantiallyabsent.

To this end the system according to the invention is characterized inthat the basis matrix C is orthogonal and in that the elements c_(2i),kof the even basis functions and the elements C_(2i+l),k (k=0, 1, . . . )of the odd basis functions c_(2i+1), for at least the lowest order (i=0)comply with:

    0≦|c.sub.2i,0 +c.sub.2i+1,0 |≦|c.sub.2i,1 +c.sub.2i+1,1 |

The invention is based on the recognition that an orthogonal lappedtransform matrix can be constructed from any arbitrary orthogonal K*Kbasis matrix C. It is achieved by means of a basis matrix complying withthe imposed requirements that at least the lowest order basis functionof the lapped transform at both edges tends to the value of zero. Sincethe output signal is a weighted sum of basis functions and the lowestorder basis function is the most significant, discontinuities are nowsubstantially absent.

It is to be noted that it is known per se to scale at least the firsteven row or the first odd row of the DCT matrix in such a way that theirdifference converges towards the value of zero. However, the lappedtransform thus obtained, referred to as Modified Lapped Transform (MLT),is no longer orthogonal.

A further embodiment of the system according to the invention ischaracterized in that the elements c_(2i),k of the even basis functionsc_(2i) and the elements c_(2i+1),k of the odd basis functions c_(2i+1)pair-wise comply with:

    0≦|c.sub.2i,0 +c.sub.2i+1,0 |≦|c.sub.2i,1 +c.sub.2i+1,1 |

It is thereby achieved that all basis functions of the lapped transformat the edges tend towards zero. Block artefacts now remain completelyabsent. Moreover, the basis functions of the lapped transform have anincreasing order. At a suitable choice of C, this results in goodfrequency-discriminating properties. The lapped transform is thenextremely suitable for a substantial compression of picture and soundsignals.

A particularly favourable embodiment of the system is characterized inthat all elements c₀,k of the even basis function c₀ of the lowest orderhave the same value. Series of samples of equal value, for examplepicture blocks of uniform brightness, can then be transmitted with onecoefficient only.

These and other aspects of the invention will be apparent from andelucidated with reference to the embodiments described hereinafter.

BRIEF DESCRIPTION OF THE DRAWING

In the drawings:

FIGS. 1 and 2 show embodiments of a system for transmitting picture orsound signals according to the invention.

FIG. 3 shows block artefacts as occur in non-lapped transforms.

FIG. 4 shows the two first basis functions of the non-lapped DCT.

FIG. 5 shows the first two basis functions of the LOT in accordance withthe prior art.

FIG. 6 shows block artefacts as occur in the known LOT.

FIG. 7 illustrates the realisation of the two first basis functions of abasis matrix for constructing a LOT according to the invention.

FIG. 8 shows the two first basis functions of the LOT according to theinvention.

FIG. 9 shows the response of the system according to the invention to alinearly increasing input signal upon transmission of a singlecoefficient.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

FIG. 1 shows a first embodiment of a system for transmitting picture orsound signals according to the invention. The device comprises a shiftregister 1a, 1b for series-parallel conversion of successive samplesx_(k) of an input signal. The shift register comprises 2K elements andis shown in the form of two sub-registers 1a and 1b of K elements so asto indicate that the input signal after each transform of 2K samples isshifted over a length of K samples. The 2K available samples are denotedby x₁. . . x_(2K). They are applied to a transformer 2 in which thevector {x₁. . . x_(2K) } is multiplied by a K*2K transform matrix. Thismatrix will further be referred to as the analysis matrix A. Eachtransform of 2K input samples yields K coefficients y₁ . . . y_(K). Theyare applied to a quantization and coding circuit 3 and subsequentlytransmitted via a transmission medium 4. The transmission medium may bea radio or cable connection, but also a storage medium such as amagnetic tape or optical disc. At the receiver end, the quantizedcoefficients y₁ . . . y_(k) are applied to a transformer 5. Thistransformer multiplies each series {y₁ . . . y_(K) } of K coefficientsby a 2K*K matrix S¹. The matrix S^(t) is the transposed version of aK*2K matrix S which will further be referred to as the synthesis matrixS. The transform of the K coefficients yield 2K numbers z₁ . . . z_(2K).These are summed in an overlapping manner. This is shown in the Figureby means of adders 7a . . . 7k in which each time K numbers z₁ . . .z_(K) are added to K numbers z_(K+1) . . . z_(2K) from the precedingseries. To this end, the relevant part of the preceding series is storedin a register 5. The samples u₁ . . . u_(K) of the reconstructed pictureor sound signal are available at the output of adders 7a . . . 7k.

Nowadays it is realised that a signal transform may essentially beconsidered as a special case of multirate filter bank coding. Such amultirate filter bank is shown in FIG. 2 as a second embodiment of asystem according to the invention. The filter bank comprises K filters10a . . . 10k having a filter length 2K and a transfer function H_(k)(z). The filtered signals are decimated by a factor K in down-samplers11a . . . 11k. This is generally referred to as critical sampling. Itmeans that only each Kth sample of the filtered signals is transmittedwhile the intermediate K-1 samples are ignored. Each transmitted samplerepresents a coefficient y_(k) . After quantization and transmission,the K signals are up-sampled (filling in the K-1 intermediate samples ofthe value zero) in up-samplers 12a . . . 12k. Subsequently, they areapplied to K interpolation filters 13a . . . 13k having a filter length2K and a transfer function F_(k) (z). The interpolated signals aresubsequently summed in an adder 14.

The analogy between transform (FIG. 1) and multirate filter bank coding(FIG. 2) provides the possibility of gaining an insight into someaspects of the invention. For example, the K pulse responses having alength 2K of the interpolation filters F_(k) (z) correspond to the Krows of the K*2K transformer S, and the K pulse responses having alength 2K of the filters H_(K) (z) correspond to the rows of the matrixA which are reversed in order.

To understand the embodiments, signal transforms in general will beelucidated first. In principle, the picture or sound signal x to betransmitted comprises an infinite number of samples x_(k). Due to theforward transform at the transmitter end of the circuit, it is convertedinto an equally large number of coefficients y_(k). These coefficientsare backward-transformed at the receiver end. The forward and backwardtransforms are laid down in a transform matrix. The transform matrix atthe transmitter end will be denoted by T and at the receiver end by P.In a mathematical notation, the forward transform is described by y=Tx.The backward transform is described by u=P^(t) y, in which P^(t) is thetransposed version of P (transposition is the exchange of rows andcolumns). If the matrix P is written in the form of ##EQU2## then thebackward transformation ##EQU3## can be written as: ##EQU4## This vectornotation shows that the output signal {u₁,u₂ . . . } can be consideredto be a linear combination of vectors {p₁₁,p₁₂, . . . }, {p₂₁, p₂₂, . .. }, etc. with weighting factors y₁,y₂ . . . Said vectors areconstituted by the columns of P^(t), i.e. the rows of matrix P. They arereferred to as the basis functions of transformer P. The weightingfactors y₁,y₂, . . . are the coefficients obtained from the forwardtransform.

We will now concentrate on separable transforms. This means that2-dimensional transforms (for example, of video pictures) can beperformed by performing a 1-dimensional transform in the horizontaldirection and subsequently in the vertical direction. The followingdescription may therefore be limited to 1-dimensional transforms.

The matrices T and P are square and, in principle, infinitely large. Inpractical uses, for example in picture and sound coding, the inputsignal x is split up into 1-dimensional series (sound) or 2-dimensionalblocks (picture) with a limited number of samples. Each series or blockis subjected to the same transform. For example, in picture coding it isconventional practice to transform blocks of 8*8, 16*16 or 32*32 pixels.Due to this division, the matrices T and P acquire a block-Toeplitzstructure which can be mathematically written as follows: ##EQU5## inwhich A (analysis) and S (synthesis) are the matrices of finitedimensions shown in FIG. 1. The rows of P correspond to the rows of S,supplemented with zeroes on either side. The basis functions of thetransform are thus only unequal to zero in a limited range. The matricesA and S in equation (2) may or may not overlap each other. Thefrequently used DCT (inter alia, in MPEG picture coding) belongs to theclass of non-lapped transforms. Here, each time K samples {x₁ . . .x_(K) } are transformed into N coefficients {y₁ . . . y_(K) } by meansof a K*K matrix A.

For data compression of video pictures, the basis functions are chosento be such that each of them is representative of a given extent ofpicture detail. In terms of the multirate filter bank: each filter H_(k)(z) filters a limited band from the frequency spectrum. As the filterfrequency is higher, the pulse response has a higher order. Since thehuman eye is less sensitive to the higher spatial frequencies, thecorresponding coefficients can be quantized in a coarser manner. Pictureplanes of uniform brightness can even be described by means of onecoefficient.

Quantization of the coefficients implies that the input signal is nolonger transmitted faultlessly. This may be formulated mathematically byassuming that each coefficient y_(k) is beset with a quantization errore_(k). Instead of the exact coefficients y, quantized coefficients y=y+eare transmitted. The output signal u of the transmission circuit thenis:

    u=P.sup.t y=P.sup.t (y+e)=P.sup.t y+P.sup.t e=x+P.sup.t e

In other words, the reconstructed output signal u not only comprises theexact input signal x but also an error signal P^(t) e. Said error signalis also a linear combination of the basis functions of P, now withe={e₁,e₂, . . . } as weighting factors. Noticeable artefacts areproduced at high compression factors, i.e. at large values of e_(k) bycoarse quantization, or even omission of coefficients of higher orderbasis functions. In non-lapped transforms, these artefacts becomerapidly manifest at the edges of each series of samples. For example, inpicture coding, luminance discontinuities occur at the edge of eachblock of pixels. FIG. 3 shows an example of this. In this example, xrepresents an input signal in the form of a linearly increasing picturebrightness which is subjected to a 16*16 DCT. It will be readily evidentthat when omitting all coefficients, except that of the lowest order(the DC coefficient), a step-wise increasing output signal u will beobtained, whose discontinuities coincide with the edges of each seriesof pixels.

In lapped transforms, the matrices A and S overlap each other partly inthe Toeplitz structure. This overlap may be expressed by writing theToeplitz structure as: ##EQU6## in which ##EQU7## are now rectangularK*N (vertical*horizontal) matrices. A₁,A₂, . . . and S₁,S₂, . . . areK*K matrices.

Due to the lapped transform, a series of N samples {x₁ . . . x_(N) } istransformed into a series of K coefficients {y₁ . . . y_(K) }. Aftereach transform, the input signal shifts by K samples. Similarly as inthe non-lapped transform, the total number of coefficients remains equalto the total number of samples.

The backward transform u=P^(t) y can be written as: ##EQU8## The outputsignal is again a sum of weighted (and now overlapping) basis functions.The basis functions are constituted by the rows {s₁₁ . . . s_(1N) } . .. {s_(K1) . . . s_(KN) } of the matrix S, while the coefficients y againconstitute the weighting factors.

We will now search for practically usable lapped signal transforms, forexample for picture coding. To this end, a number of sensiblelimitations and requirements will be imposed on the transmission chainand these will be translated into conditions with which the transformersA and S should comply.

The following description will be based on a 50% overlap. Successiveseries of N=2K samples x thus overlap each other halfway. Each series of2K samples yields K coefficients y to be transmitted. The matrices A andS have a dimension of K*2K (vertical*horizontal) elements and can beconsidered as a succession of two K*K matrices:

    A= A.sub.1 A.sub.2 ! and S= S.sub.1 S.sub.2 !

The analysis filters H_(k) (z) and synthesis filters F_(k) (z) in FIG. 2may be required to have a linear phase variation. The delay time of thesignals through each K filter is then equal. To this end, half of thepulse responses of the filters should be even and half of them should beodd. As already stated, the pulse responses correspond to the rows of Aand S. If the even and odd rows are grouped together, then A and S bothhave the following structure: ##EQU9## This property can bemathematically written as: ##EQU10## Here, A₁₁,A₂₁,S₁₁ and S₂₁ are1/2K*K matrices, J is the counter-identity matrix whose elements on theantidiagonal have the value 1 and the other elements have the valuezero. The multiplication of a matrix by J causes mirroring of the rows.

An important requirement with respect to lapped transform is a perfectreconstruction of the input signal x. To this end it is necessary thatu=P^(t) Tx, hence P^(t) T=I. Since P and T are square matrices, it isnecessary that TP^(t) =I. Given the Toeplitz structure of P^(t) and T,TP^(t) can be written as: ##EQU11## A sufficient and necessary conditionfor perfect reconstruction therefore is:

    A.sub.1 S.sub.1.sup.t +A.sub.2 S.sub.2.sup.t =I

    A.sub.2 S.sub.1.sup.t =A.sub.1 S.sub.2.sup.t =O

Since A and S in formula (3) have the same structure, these conditionscan be written as:

    A.sub.11 S.sub.11.sup.t =A.sub.21 S.sub.21.sup.t =1/2I     (4)

    A.sub.11 JS.sub.11.sup.t =A.sub.11 JS.sub.21.sup.t =A.sub.21 JS.sub.11.sup.t =A.sub.21 JS.sub.21.sup.t =O              (5)

Since the matrices A₁₁ and A₂₁ have the same rank (1/2K), the vectors ofA₁₁ and A₂₁ should subtend the same space. The rows of A₂₁ can thereforebe written as a linear combination of the rows of A₁₁, and conversely.Mathematically, this means that A₂₁ =T_(a).A₁₁, in which T_(a) is anarbitrary invertible 1/2K*1/2K matrix. The same considerations apply tothe synthesis matrix S. Formula (3) can now be written as: ##EQU12## Bymeans of T_(a) and T_(s), the filters are formed in such a way that theyhave a desired behaviour with respect to "smoothness", frequencydiscrimination and the like. Particularly, T_(a) and T_(s) cause the oddpulse responses in the centre to pass "smoothly" through zero. They arecompletely arbitrary, provided that they are invertible. The matricesT_(a) and T_(s) are not independent of each other. It can be derivedfrom condition (5) that T_(a) T_(s) ^(t) =I, hence T_(s) =T_(a) ^(-t).

The K*2K transform matrices A and S can thus be composed from a 1/2K*Kmatrix A₁₁ and S₁₁, respectively. Similarly as each row can be writtenas the sum of an even row and an odd row by means of a Fourierdevelopment, A₁₁ and S₁₁ can be written as the sum of an even matrixA_(e) and S_(e) of even rows, and an odd matrix A_(o) and S_(o) of oddrows, respectively. Formula (6) will then be: ##EQU13## The conditions(4) and (5) for perfect reconstruction then change into:

    A.sub.e S.sub.e.sup.t =A.sub.o S.sub.o.sup.t =1/4I         (8)

The forward transform A and the backward transform S^(t) are required tobe orthogonal. The analysis matrix A and the synthesis matrix S are thenequal to each other. Both of them are expressed as follows: ##EQU14## Itfollows from the foregoing that A and S can be constructed from a 1/2K*Kmatrix C_(e) which consists of even rows and a 1/2K*K matrix C_(o) whichconsists of odd rows. In other words: a lapped orthogonal transform(LOT) can be reconstructed from an orthogonal K*K matrix C, half ofwhich consists of even rows and the other half consists of odd rows. Theeven rows of C are then incorporated in C_(e) and scaled with a factor1/2. The odd rows of C are incorporated in C_(o) and are also scaledwith a factor 1/2. The orthogonal matrix C is here referred to as the"basis matrix".

In the past, research was only done into the applicability of the DCT asa basis matrix for lapped transform. To this end, reference is made tothe article mentioned in the opening paragraph, and to "The LOT:Transform Coding Without Blocking Effects" in IEEE Trans. on ASSP, vol.37, no. 4, April 1989, pp. 553-559. It is sensible to subject the knownLOT on the basis of the DCT to a further consideration. Saidpublications use the following formula: ##EQU15## in which D_(e) andD_(o) comprise the even and odd basis functions of the DCT matrix.Formula (10) is comparable with the transposed version of formula (9).The two basis functions d₀ and d₁ of the lowest order (the lowestspatial picture frequencies in picture coding) of a 16:16 DCT matrix areshown in FIG. 4. Here, d₀ is the first even row of the DCT (the firstrow of D_(e)) and d₁ is the second row of the DCT (the first row ofD_(o)). The first basis function of the LOT formed therefrom is obtainedin accordance with formula (10) by forming the difference d₀ -d₁ andrepeating this in a mirrored way so that an even basis function of 32samples is obtained. This first basis function is denoted by s₀ in FIG.5. The first odd basis function of the known LOT is obtained by formingthe difference d₀ -d₁, performing the multiplication by elements of thematrix Z' and repeating the result antisymmetrically. It should be notedthat without multiplication by Z', discontinuities would occur in thisodd basis function. Such a discontinuous basis function is undesirable.The multiplication by Z' aims at preventing these discontinuities. Thefirst odd basis function thus obtained is denoted by s₁ in FIG. 5.

In a corresponding manner, the second even basis function s₂ and secondodd basis function s₃ of the known LOT are obtained from the third(even) row d₂ and fourth (odd) row d₃ of the DCT. These and furtherbasis functions of the known LOT are not shown in FIG. 5.

The known LOT has the property of a perfect reconstruction. Moreover, ithas the property that a series of constant input samples (for example, apicture block of constant brightness) can be transmitted by means of oneDC coefficient only (the weighting factor for basis function s₀.However, the known LOT is not free from block artefacts. FIG. 6 showsthe output signal u at a linearly increasing input signal x if only thecoefficient of the first basis function is transmitted. The artefactsare produced because the first basis function s₀ at the edges has avalue which is unequal to zero. This is caused by the fact that the"peripheral value" of the first basis function d₀ of the DCT basismatrix differs by a factor of √2 from the peripheral value of d₁. It hasbeen tried to solve this by taking s₀ =d₀ √2-d₁ instead of s₀ =d₀ -d₁ atleast for the first basis function s₀. The transform obtained thereby isreferred to as "Modified Lapped Transform" (MLT) and, in practice,appears to reduce the block artefacts to a considerable extent. However,the MLT is no longer orthogonal.

A method of obtaining a suitable orthogonal basis matrix C will now bedescribed. Due to the favourable filter properties of the DCT, the DCTmatrix will be the starting point. The rows of the DCT matrix, arrangedaccording to filter frequency, are referred to as d₀ . . . d_(K) inwhich even indices indicate even rows and odd indices indicate odd rows.The rows of the basis matrix C are accordingly denoted c₀ . . . c_(K).

The first row d₀ of the DCT is taken as the first row c₀ :c₀ =d₀. Infact, the row d₀ satisfies the desired property of all elements havingthe same value. This first row is denoted by c₀ in FIG. 7. The value ofthe elements is here √(1K) so that the norm of the row (the root of thesum of the squares) is 1.

Subsequently, c₁ is determined. This row should comply with thefollowing conditions:

The elements at the left edge should go "smoothly" to √(1/K) in orderthat the elements of c₀ +c₁ go "smoothly" to zero.

c₁ should be odd, i.e. it should have a zero crossing in the centre.

The norm of the row should be equal to that of the other rows. c₁ maylargely be dimensioned intuitively. A first approximation of -c₁ isobtained by determining an interpolation curve of a higher order bymeans of some predetermined points. One or two of these points (70 inFIG. 7) are located to the left of the row and have the value √(1/K).They cause c₁ at the left edge to converge "smoothly" to the desiredvalue √(1/K). Further points are formed by the values 72 of d₁ which arelocated on both sides of the zero crossing. The filter characteristic ofc₁ will then not be essentially different from the favourablecharacteristic of the first odd row of the DCT matrix. A monotonousinterpolation curve between the points determined so far would yield arow of the norm <1. To restore the norm, at least one point having avalue which is larger than the peripheral value √(1/K) (71 in FIG. 7) isdetermined between the left edge and the zero crossing.

An interpolation curve of a higher order is computed through the pointsthus determined. The curve obtained is a first estimation of thesearched row c₁. The norm of this row is subsequently computed. Adeviation from the desired value is corrected by correcting the obtainedvalues by an appropriate factor. If desired, the normalization may takeplace in a number of iterative steps. In this way the row denoted by -c₁in FIG. 7 is eventually obtained. For the purpose of comparison, thefirst odd row of the DCT matrix is denoted by d₁ in this Figure.

The further even rows c₂,c₄, . . . of the basis matrix C may becompletely identical to the corresponding even rows d₂,d₄, . . . of theDCT matrix. They are already perpendicular to each other (because thatis a property of the DCT) and perpendicular to all other rows (becausethey are odd rows).

The further odd rows c₃,c₅, . . . of the basis matrix C may also bedirectly derived from the corresponding rows of the DCT matrix. Theirshape need not be changed and they sufficiently comply with therequirement that they should be equal at the left edge to their adjacenteven row. The rows d₃,d₅, . . . of the DCT matrix are, however, notperpendicular to the row c₁ just constructed. Consequently, they must besuccessively rotated. This process of vector rotation is generally knownin mathematics by the name of Gram-Schmidt orthogonalization.

FIG. 8 shows the first even basis function s₀ and the first odd basisfunction s₁ of the transform matrix S as obtained from c₀ and c₁ bymeans of formula (9). FIG. 9 shows the output signal u at a linearlyincreasing input signal x if only the coefficient of the first basisfunction is transmitted. Block artefacts at the edges are now completelyabsent.

What is claimed is:
 1. A system for transmitting picture or soundsignals, comprising:first transform means (2; 10, 11) for transformingseries of samples of the picture or sound signal overlapping each otherhalfway into coefficients by means of a first transform matrix A; means(3, 4) for transmitting the coefficients; second transform means (5; 12,13) for lapped backward transform of the transmitted coefficients intoan output signal by means of a second transform matrix S, in which thematrices A and S can be written as: ##EQU16## in which I is the unitmatrix, O is the zero matrix and J is the counter-identity matrix, andin which the matrices C_(e) and C_(o) comprise even basis functions andodd basis functions, respectively, of a basis matrix C, each basisfunction c_(j) having an order j which is arranged in accordance withthe central frequency of the frequency spectrum of the basis function,characterized in that the basis matrix C is orthogonal and in that theelements c_(2i),k of the even basis functions c_(2i) and the elementsc_(2i+1),k of the odd basis functions c_(2i+1) for at least the lowestorder (i=0) comply with:

    0≦|c.sub.2i,0 +c.sub.2i+1,0 |≦|c.sub.2i,1 +c.sub.2i+1,1 |.


2. A system as claimed in claim 1, characterized in that the elementsc_(2i),k of the even basis functions c_(2i) and the elements c_(2i+1),kof the odd basis functions c_(2i+1) pair-wise comply with:

    0≦|c.sub.2i,0 +c.sub.2i+1,0 |≦|c.sub.2i,1 +c.sub.2i+1,1 |.


3. A system as claimed in claim 1, characterized in that all elementsc₀,k of the even basis function c₀ of the lowest order have the samevalue.
 4. A system as claimed in claim 1, characterized in that at leastone element c₁,k (71) of the odd basis function c₁ of the lowest ordercomplies with:

    |c.sub.1,k |>|c.sub.1,0 |.


5. A coding station for coding picture or sound signals, comprisingtransform means for transforming series of samples of the picture orsound signal overlapping each other halfway into coefficients by meansof a first transform matrix A, and means for coding the coefficients, inwhich the matrix A can be written as ##EQU17## in which I is the unitmatrix, O is the zero matrix and J is the counter-identity matrix, andin which the matrices C_(e) and C_(o) comprise even basis functions andodd basis functions, respectively, of a basis matrix C, each basisfunction c_(j) having an order j which is arranged in accordance withthe central frequency of the frequency spectrum of the basisfunction,characterized in that the basis matrix C is orthogonal and inthat the elements c_(2i),k of the even basis functions c_(2i) and theelements c_(2i+1),k of the odd basis functions c_(2i+1) for at least thelowest order (i=0) comply with:

    0≦|c.sub.2i,0 +c.sub.2i+1,0 |≦|c.sub.2i,1 +c.sub.2i+1,1 |.


6. A coding station as claimed in claim 5, characterized in that theelements c_(2i),k of the even basis functions c_(2i) and the elementsc_(2i+1),k of the odd basis functions c_(2i+1) pair-wise comply with:

    0≦|c.sub.2i,0 +c.sub.2i+1,0 |≦|c.sub.2i,1 +c.sub.2i+1,1 |.


7. A coding station as claimed in claim 5, characterized in that allelements c₀,k of the even basis function c₀ of the lowest order have thesame value.
 8. A system as claimed in claim 5, characterized in that atleast one element c₁,k (71) of the odd basis function c₁ of the lowestorder complies with:

    |c.sub.1,k |>|c.sub.1,0 |.


9. A decoding station for decoding picture or sound signals transmittedin the form of coefficients of a lapped signal transform, comprisingtransform means for lapped backward transform of the transmittedcoefficients into an output signal by means of a transform matrix S^(t),in which the matrix S can be written as ##EQU18## in which I is the unitmatrix, O is the zero matrix and J is the counter-identity matrix, andin which the matrices C_(e) and C_(o) comprise even basis functions andodd basis functions, respectively, of a basis matrix C, each basisfunction c_(j) having an order j which is arranged in accordance withthe central frequency of the frequency spectrum of the basisfunction,characterized in that the basis matrix C is orthogonal and inthat the elements c_(2i),k of the even basis functions c_(2i) and theelements c_(2i+1),k of the odd basis functions c_(2i+1) for at least thelowest order (i=0) comply with:

    0≦|c.sub.2i,0 +c.sub.2i+1,0 |≦|c.sub.2i,1 +c.sub.2i+1,1 |.


10. A decoding station as claimed in claim 9, characterized in that theelements c_(2i),k of the even basis functions c_(2i) and the elementsc_(2i+1),k of the odd basis functions c_(2i+1) pair-wise comply with:

    0≦|c.sub.2i,0 +c.sub.2i+1,0 |≦|c.sub.2i,1 +c.sub.2i+1,1 |.


11. A decoding station as claimed in claim 9, characterized in that allelements c₀,k of the even basis function c₀ of the lowest order have thesame value.
 12. A decoding station as claimed in claim 9, characterizedin that at least one element c₁,k (71) of the odd basis function c₁ ofthe lowest order complies with:

    |c.sub.1,k |>|c.sub.1,0 |.


13. A method of transmitting picture or sound signals, comprising thesteps of:transforming series of samples of the picture or sound signaloverlapping each other halfway into coefficients by means of a firsttransform matrix A; transmitting the coefficients; lapped backwardtransform of the transmitted coefficients into an output signal by meansof a second transform matrix S^(t), in which the matrices A and S can bewritten as ##EQU19## in which I is the unit matrix, O is the zero matrixand J is the counter-identity matrix, and in which the matrices C_(e)and C_(o) comprise even basis functions and odd basis functions,respectively, of a basis matrix C, each basis function c_(j) having anorder j which is arranged in accordance with the central frequency ofthe frequency spectrum of the basis function, characterized in that thebasis matrix C is orthogonal and in that the elements c_(2i),k of theeven basis functions c_(2i) and the elements c_(2i+1),k of the odd basisfunctions c_(2i+1) for at least the lowest order (i=0) comply with:

    0≦|c.sub.2i,0 +c.sub.2i+1,0 |≦|c.sub.2i,1 +c.sub.2i+1,1 |.


14. A method as claimed in claim 13, characterized in that the elementsc_(2i),k of the even basis functions c_(2i) and the elements c_(2i+1),kof the odd basis functions c_(2i+1) pairwise comply with:

    0≦|c.sub.2i,0 +c.sub.2i+1,0 |≦|c.sub.2i,1 +c.sub.2i+1,1 |.


15. A method as claimed in claim 13, characterized in that all elementsc₀,k of the even basis function c₀ of the lowest order have the samevalue.
 16. A method as claimed in claim 13, characterized in that atleast one element c₁,k (71) of the odd basis function c₁ of the lowestorder complies with:

    |c.sub.1,k |>|c.sub.1,0 |.